Bifurcation Theory and Applications

✍ Scribed by Tian Ma; Shouhong Wang

• Publisher: World Scientific
• Year: 2005
• Tongue: English
• Leaves: 391
• Category: Library

✦ Synopsis

• Provides a comprehensive and intuitive review of existing bifurcation theories - New theories for bifurcations from eigenvalues with even multiplicity - General recipes for applications

✦ Table of Contents

Cover
Title
Preface
Contents
Chapter 1 Introduction to Steady State Bifurcation Theory
1.1 Implicit Function Theorem
1.2 Basics of Topological Degree Theory
1.2.1 Brouwer degree
1.2.2 Basic theorems of Brouwer degree
1.2.3 Leray-Schauder degree
1.2.4 Indices of isolated singularities
1.3 Lyapunov-Schmidt Method
1.3.1 Preliminaries
1.3.2 Lyapunov–Schmidt procedure
1.3.3 Normalization
1.4 Krasnosel’ski Bifurcation Theorems
1.4.1 Bifurcation from eigenvalues with odd multiplicity
1.4.2 Krasnosel’ski theorem for potential operators
1.5 Rabinowitz Global Bifurcation Theorem
1.6 Notes
Chapter 2 Introduction to Dynamic Bifurcation
2.1 Motivation
2.2 Semi-groups of Linear Operators
2.2.1 Introduction
2.2.2 Strongly continuous semi-groups
2.2.3 Sectorial operators and analytic semi-groups
2.2.4 Powers of linear operators
2.3 Dissipative Dynamical Systems
2.4 Center Manifold Theorems
2.4.1 Center and stable manifolds in Rn
2.4.2 Center manifolds for infinite dimensional systems
2.4.3 Construction of center manifolds
2.5 Hopf Bifurcation
2.6 Notes
Chapter 3 Reduction Procedures and Stability
3.1 Spectrum Theory of Linear Completely Continuous Fields
3.1.1 Eigenvalues of linear completely continuous fields
3.1.2 Spectral theorems
3.1.3 Asymptotic properties of eigenvalues
3.1.4 Generic properties
3.2 Reduction Methods
3.2.1 Reduction procedures
3.2.2 Morse index of nondegenerate singular points
3.3 Asymptotic Stability at Critical States
3.3.1 Introduction to the Lyapunov stability
3.3.2 Finite dimensional cases
3.3.3 An alternative principle for stability
3.3.4 Dimension reduction
3.4 Notes
Chapter 4 Steady State Bifurcations
4.1 Bifurcations from Higher Order Nondegenerate Singularities
4.1.1 Even–order nondegenerate singularities
4.1.2 Bifurcation at geometric simple eigenvalues: r = 1
4.1.3 Bifurcation with r=k=2
4.1.4 Reduction to potential operators
4.2 Alternative Method
4.2.1 Introduction
4.2.2 Alternative bifurcation theorems
4.2.3 General principle
4.3 Bifurcation from Homogeneous Terms
4.4 Notes
Chapter 5 Dynamic Bifurcation Theory: Finite Dimensional Case
5.1 Introduction
5.1.1 Pendulum in a symmetric magnetic field
5.1.2 Business cycles for Kaldor’s model
5.1.3 Basic principle of attractor bifurcation
5.2 Attractor Bifurcation
5.2.1 Main theorems
5.2.2 Stability of attractors
5.2.3 Proof of Theorems 5.2 and 5.3
5.2.4 Structure of bifurcated attractors
5.2.5 Generalized Hopf bifurcation
5.3 Invariant Closed Manifolds
5.3.1 Hyperbolic invariant manifolds
5.3.2 S1 attractor bifurcation
5.4 Stability of Dynamic Bifurcation
5.5 Notes
Chapter 6 Dynamic Bifurcation Theory: Infinite Dimensional Case
6.1 Attractor Bifurcation
6.1.1 Equations with first-order in time
6.1.2 Equations with second-order in time
6.2 Bifurcation from Simple Eigenvalues
6.2.1 Structure of dynamic bifurcation
6.2.2 Saddle-node bifurcation
6.3 Bifurcation from Eigenvalues with Multiplicity Two
6.3.1 An index formula
6.3.2 Main theorems
6.3.3 Proof of main theorems
6.3.4 Case where k 3
6.3.5 Bifurcation to periodic solutions
6.4 Stability for Perturbed Systems
6.4.1 General case
6.4.2 Perturbation at simple eigenvalues
6.5 Notes
Chapter 7 Bifurcations for Nonlinear Elliptic Equations
7.1 Preliminaries
7.1.1 Sobolev spaces
7.1.2 Regularity estimates
7.1.3 Maximum principle
7.2 Bifurcation of Semilinear Elliptic Equations
7.2.1 Transcritical bifurcations
7.2.2 Saddle-node bifurcation
7.3 Bifurcation from Homogenous Terms
7.3.1 Superlinear case
7.3.2 Sublinear case
7.4 Bifurcation of Positive Solutions of Second Order Elliptic Equations
7.4.1 Bifurcation in exponent parameter
7.4.2 Local bifurcation
7.4.3 Global bifurcation from the sublinear terms
7.4.4 Global bifurcation from the linear terms.
7.5 Notes
Chapter 8 Reaction-Diffusion Equations
8.1 Introduction
8.1.1 Equations and their mathematical setting
8.1.2 Examples from Physics, Chemistry and Biology
8.2 Bifurcation of Reaction-Diffusion Systems
8.2.1 Periodic solutions
8.2.2 Attractor bifurcation
8.3 Singularity Sphere in Sm-Attractors
8.3.1 Dirichlet boundary condition
8.3.2 Periodic boundary condition
8.3.3 Invariant homological spheres
8.4 Belousov-Zhabotinsky Reaction Equations
8.4.1 Set-up
8.4.2 Bifurcated attractor
8.5 Notes
Chapter 9 Pattern Formation and Wave Equations
9.1 Kuramoto-Sivashinsky Equation
9.1.1 Set-up
9.1.2 Symmetric case
9.1.3 General case
9.1.4 S1-invariant sets
9.2 Cahn-Hillard Equation
9.2.1 Set-up
9.2.2 Neumann boundary condition
9.2.3 Periodic boundary condition
9.2.4 Saddle-node bifurcation
9.3 Complex Ginzburg-Landau Equation
9.3.1 Set-up
9.3.2 Dirichlet boundary condition
9.3.3 Periodic boundary condition
9.4 Ginzburg-Landau Equations of Superconductivity
9.4.1 The model
9.4.2 Attractor bifurcation
9.4.3 Physical remarks
9.5 Wave Equations
9.5.1 Wave equations with damping
9.5.2 System of wave equations
9.6 Notes
Chapter 10 Fluid Dynamics
10.1 Geometric Theory for 2-D Incompressible Flows
10.1.1 Introduction and preliminaries
10.1.2 Structural stability theorems
10.2 Rayleigh-Benard Convection
10.2.1 Benard problem
10.2.2 Boussinesq equations
10.2.3 Attractor bifurcation of the Rayleigh-Benard problem
10.2.4 2-D Rayleigh-Benard convection
10.3 Taylor Problem
10.3.1 Taylor’s experiments and Taylor vortices
10.3.2 Governing equations
10.3.3 Stability of secondary flows
10.3.4 Taylor vortices
10.4 Notes
Bibliography
Index
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